Arrowroot Ash should appear in the Alchemy Shop of the Dragon Mountains.

To free Lumenne - re are secret switches you have to touch in Sanctum of the Ice Trolls.

Stellar Abyss - For the Stellar Abyss, facing losing all your hp at the statue, you need the Amulet of Joining. You have to go to Lord Orion to get that but make sure you've talked to the Asteroid Mage first and listened to her instructions.

"The Mages await you in the Temple of the Great Mage, just inside Mage HQ. If you bring each of them their Temple key, each will give you a scroll. Place those scrolls on the stand there. That will summon the Light Mage.

He will ask you to go to Lord Orion and get an amulet that will allow you to lure the Dark Mage to him. If you can do that, the Great Mage will be restored and our long battle will be over. Then see the Elemental Spirit. If you've found all the pieces of your ship, you can go home."